# Elimination ideals and Bezout relations

## TL;DR

This paper investigates bounds on the degrees of elimination ideals and Bezout relations for polynomial ideals over infinite fields, focusing on the minimal degrees of polynomials depending on a subset of variables.

## Contribution

It provides new bounds on the minimal degrees of polynomials in elimination ideals and Bezout relations, advancing understanding of polynomial ideal structure.

## Key findings

- Established bounds on degrees of elimination polynomials.
- Derived bounds for degrees of Bezout relations.
- Enhanced understanding of polynomial ideal generators.

## Abstract

Let $k$ be an infinite field and $I\subset k [x_1, \ldots ,x_n]$ be an ideal such that dim $V(I)=q$. Denote by $(f_1, \ldots, f_s)$ a set of generators of $I$. One can see that in the set $I\cap k [x_{1},...,x_{q+1}]$ there exist non-zero polynomials, depending only on these $q+1$ variables. We aim to bound the minimal degree of the polynomials of this type, and of a B\'ezout (i.e. membership) relation expressing such a polynomial as a combination of the $f_i$.